where the first and second terms represent the contributions associated
with the \(gg \rightarrow c \bar{c}\)  and \( cg \rightarrow cg \)  mechanisms,
with the corresponding expressions depending on the factorization
scheme assumed in the calculations. In
Ref. [1]}, a detailed comparison between the
collinear, hybrid and \(k_T\) -factorization approaches was performed.
In what follows, we will focus on the hybrid factorization model, which
is based on the studies performed also in
Refs. [2]}, [3]}, [4]}, [5]}.
Such a choice is motivated by:
(a) the theoretical expectation that the
collinear approach, largely used in previous calculations of
\(\phi _{\nu }\) , breaks down at very small-\(x\)  [3]}, [5]};
and that (b) the \(k_T\) -factorization approach reduces to the hybrid
model in the dilute-dense regime, which is the case in the charm
production at very forward rapidities, where we are probing large
(small) values of \(x\)  in the projectile (target).
In this approach, the differential cross sections for \(gg^* \rightarrow c\bar{c}\)  and \(cg^* \rightarrow cg\)  mechanisms,
sketched in Fig. REF , are given by
\(d \sigma _{pp \rightarrow charm}(gg \rightarrow c \bar{c}) = \int dx_1 \int \frac{dx_2}{x_2} \int d^2k_t \, g(x_1,\mu ^2) \, {\cal {F}}_{g^*} (x_2, k_t^2, \mu ^2) \, d\hat{\sigma }_{gg^* \rightarrow c\bar{c}}\) 
